Divergence

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For other uses, see Divergence (disambiguation).

In vector calculus, the divergence is an operator that measures a vector field's tendency to originate from or converge upon a given point. For instance, for a vector field that denotes the velocity of air expanding as it is heated, the divergence of the velocity field would have a positive value because the air is expanding. Conversely, if the air is cooling and contracting, the divergence would be negative.

A vector field which has zero divergence everywhere is called solenoidal.

1 Definition
2 Physical interpretation
3 Properties
4 Relation with the exterior derivative
5 Generalizations
6 See also
7 External links

[edit] Definition

Let x, y, z be a system of Cartesian coordinates on a 3-dimensional Euclidean space, and let i, j, k be the corresponding basis of unit vectors.

The divergence of a continuously differentiable vector field F = F₁ i + F₂ j + F₃ k is defined to be the scalar-valued function:

$\operatorname{div}\,\mathbf{F} = \nabla\cdot\mathbf{F} =\frac{\partial F_1}{\partial x} +\frac{\partial F_2}{\partial y} +\frac{\partial F_3}{\partial z}.$

Although expressed in terms of coordinates, the result is invariant under orthogonal transformations, as the physical interpretation suggests.

The common notation for the divergence ∇·F is a convenient mnemonic, and an abuse of notation, where the dot denotes something just reminiscent of the dot product: take the components of ∇ (see del), apply them to the components of F, and sum the results.

[edit] Physical interpretation

In physical terms, the divergence of a three dimensional vector field is the extent to which the vector field flow behaves like a source or a sink at a given point. Indeed, an alternative, but logically equivalent definition, gives the divergence as the derivative of the net flow of the vector field across the surface of a small sphere relative to the volume of the sphere. (Note that we are imagining the vector field to be like the velocity vector field of a fluid (in motion) when we use the terms flow, sink and so on.) Formally,

$( \operatorname{div}\,\mathbf{F}) (p) = \lim_{r \rightarrow 0} \int_{S(r)} {\mathbf{F}\cdot\mathbf{n}dS \over \frac{4}{3} \pi r^3 }$

where S(r) denotes the sphere of radius r about a point p in R³, and the integral is a surface integral taken with respect to n, the normal to that sphere.

In light of the physical interpretation, a vector field with constant zero divergence is called incompressible – in this case, no net flow can occur across any closed surface.

The intuition that the sum of all sources minus the sum of all sinks should give the net flow outwards of a region is made precise by the divergence theorem.

[edit] Properties

The following properties can all be derived from the ordinary differentiation rules of calculus. Most important of which, the divergence is a linear operator, i.e.

$\operatorname{div}( a\mathbf{F} + b\mathbf{G} ) = a\;\operatorname{div}( \mathbf{F} ) + b\;\operatorname{div}( \mathbf{G} )$

for all vector fields F and G and all real numbers a and b.

There is a product rule of the following type: if φ is a scalar valued function and F is a vector field, then

$\operatorname{div}(\varphi \mathbf{F}) = \operatorname{grad}(\varphi) \cdot \mathbf{F} + \varphi \;\operatorname{div}(\mathbf{F}),$

or in more suggestive notation

$\nabla\cdot(\varphi \mathbf{F}) = (\nabla\varphi) \cdot \mathbf{F} + \varphi \;(\nabla\cdot\mathbf{F}).$

Another product rule for the cross product of two vector fields F and G in three dimensions involves the curl and reads as follows:

$\operatorname{div}(\mathbf{F}\times\mathbf{G}) = \operatorname{curl}(\mathbf{F})\cdot\mathbf{G} \;-\; \mathbf{F} \cdot \operatorname{curl}(\mathbf{G}),$

$\nabla\cdot(\mathbf{F}\times\mathbf{G}) = (\nabla\times\mathbf{F})\cdot\mathbf{G} - \mathbf{F}\cdot(\nabla\times\mathbf{G}).$

The Laplacian of a scalar field is the divergence of the field's gradient.

The divergence of the curl of any vector field (in three dimensions) is constant and equal to zero. Conversely, if you have a vector field F with zero divergence defined on a ball in R³, say, then there exists some vector field G on the ball with F = curl(G). For regions in R³ more complicated than balls, this latter statement might not be true anymore (see Poincaré lemma). Indeed, the degree of failure of the truth of the statement, measured by the homology of the chain complex

$\{\mbox{scalar fields on }U\} \;$

$\to\{\mbox{vector fields on }U\} \;$

$\to\{\mbox{scalar fields on }U\} \;$

(where the first map is the gradient, the second is the curl, the third is the divergence) serves as a nice quantification of the complicatedness of the underlying region U. These are the beginnings and main motivations of de Rham cohomology.

[edit] Relation with the exterior derivative

One can establish a parallel between the divergence and a particular case of the exterior derivative, when it takes a 2-form to a 3-form in R³. If we define:

$\alpha=F_1\ dy\wedge dz + F_2\ dz\wedge dx + F_3\ dx\wedge dy$

its exterior derivative $d α$ is given by

$d\alpha = \left( \frac{\partial F_1}{\partial x} +\frac{\partial F_2}{\partial y} +\frac{\partial F_3}{\partial z} \right) dx\wedge dy\wedge dz$

[edit] Generalizations

The divergence of a vector field can be defined in any number of dimensions. If

$\mathbf{F}=(F_1, F_2, \dots, F_n),$

define

$\operatorname{div}\,\mathbf{F} = \nabla\cdot\mathbf{F} =\frac{\partial F_1}{\partial x_1} +\frac{\partial F_2}{\partial x_2}+\cdots +\frac{\partial F_n}{\partial x_n}.$

For any n, the divergence is a linear operator, and it satisfies the "product rule"

$\nabla\cdot(\varphi \mathbf{F}) = (\nabla\varphi) \cdot \mathbf{F} + \varphi \;(\nabla\cdot\mathbf{F}).$

for any scalar-valued function φ.

The divergence can be defined on any manifold of dimension n with a volume form (or density) $μ$ e.g. a Riemannian or Lorentzian manifold. Generalising the construction of a two form for a vectorfield on $\mathbb{R}^3$ , on such a manifold a vectorfield X defines a n-1 form $j = i X μ$ obtained by contracting X with $μ$ . The divergence is then the function defined by

$d j = \operatorname{div}(X) \mu$

Standard formulas for the Lie derivative allow us to reformulate this as

$\mathcal{L}_X \mu = \operatorname{div}(X) \mu$

This means that the divergence measures the rate of expansion of a volume element as we let it flow with the vectorfield.

On a Riemannian or Lorentzian manifold the divergence with respect to the metric volume form can be computed in terms of the Levi Civita connection $\nabla$

$\operatorname{div}(X) = \nabla\cdot X = X^a_{;a}$

where the second expression is the contraction of the vectorfield valued 1 -form $\nabla X$ with itself and the last expression is the traditional coordinate expression used by physicists.